Is anti-racist math a valid approach to teaching mathematics?

[kleinwolf]

This remembers me on how I got scold at during a religion class :

we got told about the multiplication of bread by Jesus :

I asked : "So he did multiply the bread by dividing it into smaller breads ?"

I got no answer but the teacher scold me : "You have to respect Jesus and what he did"...so that now I'm really doubting some physical laws about conservation...

See how other topics can really influence science ?

Another thing is : Scientist always laugh at alchemists trying to transform Lead in Gold ? Note that both are matter.

But how about your boss ? He is in fact transforming time (the one you work) in money, which is most of the time paper or some metal...So your boss is infact transforming an abstract physical reality into matter...

Isn't that a miracle ?

 

[arildno]

owl:
You are confusing math pedagogics with maths.
Learn the difference.

Secondly, no one can deny that scientists can be racists, and thus, by their personal feelings make the study/practice of science obnoxious to the targets of the scientist's prejudices.

This however, doesn't make statements like "math is racist" correct.

 

[bombadillo]

In point of fact, algebra was NOT invented by Moslems. The Babylonians, Greeks, Chinese and, most specifically, the Hindus, for examples, had algebra long before the Moslems. In fact, Moslem development of algebra is mostly based upon the work of the Hindu mathematician, Brahmagupta.

True. And as far as I know, the moslems didn't do much with developing the subject anyway. I think Omar Khayyam worked a bit with cubic equations, maybe one or two other moslem mathematicians did some small pieces of original work. Nothing like the colossal European legacy. We have to remember the moslems primarily as transmitters rather than originators. Hindus, on the other hand, were more prolific.

As for the essential equality in math and science being touted herein, if it were really true, white males would not make so much more money in equivalent positions for equivalent work that they, in fact, do. The bias against non-white males is a strong reality still in the maths and sciences. It rears up in tenure considerations, salary, available clerical support, funding, availability of research assistants, assistance getting published, and on and on.

True.

As for lower scores in math and sciences for other cultures, there are a slew of reasons why these scores exist. Some of them, are indeed, because of the way math and science are taught. However, much of this teaching is not a deliberate attempt to prevent peoples from attaining competence. It is because we are new to multicultural teaching. I have been guilty of it myself, when teaching college algebra at a D.C. university, which had students from around the world.

What about differences between the races in aptitude and inclination?

 

[cragwolf]

'When the INTELLECTUAL IMPOSTERS published in France at the end of 1997, It sent shock waves through the Left Bank establishment. Alan Sokal and Jean Bricmont showed how some of Frances's Most renowned contemporary thinkers have repeatedly abused scientific concepts in their work'. (From the back cover of the 'Intellectual Impostures' by Alan Sokal and Jean Bricmont 1997)

INTELLECTUAL IMPOSTURES should be required reading for anyone who wants to understand Post-modernism and why it is so absurd.

Yes and this sort of thing can't happen in Physics. I mean, there's no way physicists could be so "absurd". *cough* *cough*

 

[sd01g]

Yes and this sort of thing can't happen in Physics. I mean, there's no way physicists could be so "absurd". *cough* *cough*

Are you sure you read the right book? You did read it-right?

 

[honestrosewater]

I don't have exact facts that these schools are eligious schools,

So what were you basing your claim on?

but if you don't want to accept the obvious-that religion IS racist, classist, descriminational...and it affects other things to be like this-then do it, but you will be neglecting the truth. This is getting off the topic, so please don't comment on this in this thread.

You're the one getting off topic with this irrelevant slander. You claimed that most schools offering same-sex classes were religious schools, but you haven't produced any facts or reasoning to support your claim.

 

[<<<GUILLE>>>]

So what were you basing your claim on?
You're the one getting off topic with this irrelevant slander. You claimed that most schools offering same-sex classes were religious schools, but you haven't produced any facts or reasoning to support your claim.

The post you have quoted me explained:

I don't have facts, but it is obvious, just because religious school ahave ussually same-sex classes. When I postd it I based in:
1) I was going to a religious school a few years ago, and it was same-sex completeley.
2) religion has always separated same-sex. Religion has always descriminated.

But thi shas nothing to do with the original topic, so please don't carry-on with this.

Back to those posts up their, by someone I don't remember the username,

I didn't know that algebra wasn't muslim, but you see, it's hindu! So imy defend still works with this: there is no racism or descrimination of any type inside science and math, because they are based on logic and nature. The only possible racism in science is the racism of scientists.

 

[cragwolf]

Are you sure you read the right book? You did read it-right?

Yes, and I did more than read the back cover. There are two things that are absurd about what you wrote:

1. The idea that Sokal and Bricmont set out to prove that postmodernism is absurd. They attempt no such thing. In their own words, "what we intend to show" is:

The goal of this book is to make a limited but original contribution toward the critique of the admittedly nebulous Zeitgeist that we have called "postmodernism". We make no claim to analyze postmodernist thought in general; rather, our aim is to draw attention to a relatively little-known aspect, namely the repeated abuse of concepts and terminology coming from mathematics and physics. We shall also analyze certain confusions of thought that are frequent in postmodernist writings and that bear on either the content or the philosophy of the natural sciences.

2. The idea that Sokal and Bricmont actually do prove that postmodernism is absurd, even if this wasn't their original intention. This is something that narrow-minded people have extracted from the book and the affair that started it all. Fortunately, not every physicist, mathematician and http://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/sokalhoax.html bought the narrow-minded view.

3. The implied idea that physicists and practitioners from other fields of knowledge (scientific or otherwise) do not also engage in these sorts of "abuse of concepts" when they attempt to cross over into fields well outside their expertise. This is a truly laughable idea when you realize that physicists are at least equal to postmodernists in this regard. The amount of philosophical fallacies and nonsense that has spewed from the pens of such legends as Einstein, Bohr, and Heisenberg ... and later, Penrose, Weinberg, and Hawking, rivals anything that the postmodernists have written about science.

The lesson of the Sokal affair is not the infantile idea that postmodernism is absurd. Every field has problems with the peer review process. The lesson you should learn can be found, unsurprisingly, in a quote by the no-nonsense Richard Feynman:

I believe that a scientist looking at nonscientific problems is just as dumb as the next guy.

Replace "scientist" and "scientific" by their equivalents from any other field of knowledge you like and the quote would still ring true.

 

[honestrosewater]

I don't have facts, but it is obvious, just because religious school ahave ussually same-sex classes. When I postd it I based in:
1) I was going to a religious school a few years ago, and it was same-sex completeley.
2) religion has always separated same-sex. Religion has always descriminated.

You can google coed religious schools to see that this is false. The rest are non sequiturs. One religious school is same-sex, therefore most religious schools are same-sex - Invalid. Most religious schools are same-sex schools, therefore most same-sex schools are religious schools - Invalid. Besides, in this case, the reason for separating sexes is to avoid discrimination.
Also, you're arguing that people must distinguish between the tenets of science and the practice of science, so are you basing your claims on the tenets of religions or on the practice of religions? Do you know the tenets of every religion?

I'm not sure what is being proposed here. That truth and proof are defined with respect to each theory? That theories have a definite scope, i.e., apply to a certain set of objects or phenomena? That different theories with the same scope can be compatible or incompatible?

 

[sd01g]

Good work cragwolf. You did read at least to page 3 of the introduction. When you get to pages 193 and 194 you will find:

'Postmodernism has three principal negative effects: a waste of time in human sciences, a cultural confusion that favors obscurantism, and a weakening of the political left.' 'What is worse, in our opinion is the adverse effect that abandoning clear thinking and clear writing has on teaching and culture'.

The goal is not to tell anyone what to think or believe, but to encourage obtaining enough information so one can determine the absurdity of postmodernism for oneself.

 

[cragwolf]

Good work cragwolf. You did read at least to page 3 of the introduction. When you get to pages 193 and 194 you will find:

Sarcasm is only effective when you are sure about the truth of the topic in question. If you are sure that I haven't read the book then you're being an idiot and the subsequent sarcasm only makes it worse.

'Postmodernism has three principal negative effects: a waste of time in human sciences, a cultural confusion that favors obscurantism, and a weakening of the political left.' 'What is worse, in our opinion is the adverse effect that abandoning clear thinking and clear writing has on teaching and culture'.

Which in no way contradicts what I wrote. This paragraph (and the rest of the book for that matter) does not show that Sokal and Bricmont set out to prove that postmodernism is absurd. It also, for that matter, does not show that postmodernism is absurd.

The goal is not to tell anyone what to think or believe, but to encourage obtaining enough information so one can determine the absurdity of postmodernism for oneself.

Which would make your following words...

...Post-modernism and why it is so absurd.

...a clear-cut example of hypocrisy. If you simply had said, "read this book and make up your mind about postmodernism", then perhaps you would have been on reasonably steady ground. But on second thought, the idea that one book, whose authors clearly admit is only a "limited ... critique", will be sufficient to make one's mind up about an entire field of knowledge is ... absurd.

So, here's an idea. If you're actually interested in learning about postmodernism, why not read some more books on the subject? Get more than one viewpoint, perhaps? First, a nice essay on the subject:

http://www.galilean-library.org/int12.html

Second, an excellent textbook:

"Continental Philosophy: A Contemporary Introduction" by Andrew Cutrofello

And third, a web page full of links to sites of varying quality:

http://pegasus.cc.ucf.edu/~janzb/continental/

 

[arildno]

I think there is a lot of talking beside each other here:

If there are anyone here who thinks:
Maths is racist in itself (which was the issue in this thread), "Principia Mathematica" is a manual for rape (from L. Irigaray) or who holds that the imaginary unit is a phallic symbol (from J. Lacan), please do speak up.

For others, please distance yourselves from such idiotic and absurd utterances.

 

[sd01g]

If only one persons reads and understands INTELLECTUAL IMPOSTURES, after learning about it in this thread, then this thread will have provided enlightenment to at least one.

Note to cragwolf: I really liked your statement--'Sarcasm is only effective when you are sure about the truth of the topic in question.' Does this mean you believe TRUTH actually exists?

 

[<<<GUILLE>>>]

I think there is a lot of talking beside each other here:

If there are anyone here who thinks:
Maths is racist in itself (which was the issue in this thread), "Principia Mathematica" is a manual for rape (from L. Irigaray) or who holds that the imaginary unit is a phallic symbol (from J. Lacan), please do speak up.

For others, please distance yourselves from such idiotic and absurd utterances.

what's meant with imaginary unit and phalic symbol?

And, no: math itself is not racist. this is impossible.

 

[houserichichi]

I'm very sorry but I don't understand HOW math can be 'racist'. Perhaps professions therein as one poster had mentioned, but the actual subject itself has no preference to anyone; it's just abstract thought applied to abstract concepts. Perhaps those who claim so are using a bastardized definition of the word 'racist' in the first place.

Maybe I'm just confused by the whole thing.

Maybe I'm right

 

[<<<GUILLE>>>]

I'm very sorry but I don't understand HOW math can be 'racist'. Perhaps professions therein as one poster had mentioned, but the actual subject itself has no preference to anyone; it's just abstract thought applied to abstract concepts. Perhaps those who claim so are using a bastardized definition of the word 'racist' in the first place.

Maybe I'm just confused by the whole thing.

Maybe I'm right

yo are right: math and science can't be racist. Some think that the subject that should be talked about is racism of the mathematiciasns and scientists, but I think that's not the subject because that is sort of clear to many people. What I think that should be talked about here is the racism of the math application: do we use western methods in math only? do we use math from other cultures?...

I think that, yes: because algebra is hindu (I thought it was muslim, but no), our numbers are arabic...and there are many other parts of math invented by people of very different cultures: china, india, maybe africa, native americans...

 

[arildno]

what's meant with imaginary unit and phalic symbol?

Basically, Lacan equates "the exuberance of wild imagination" (his interpretation of the imaginary unit) with the wild, unfettered joy as symbolized by the phallus.

I know, it IS stupid; one of the few utterances which tops it is L. Irigaray's opinion that the main reason why the mechanics of solids (study of beams and rigid constructions and the like) has been more successful than fluid mechanics, is that beams are a phallic, masculine symbol, whereas fluids as in menstrual blood, is a symbol of the feminine.

Thus, due to the evils of patriarchy, the science of fluid mechanics has been held down by the science of solids.

 

[owl3951]

owl:
You are confusing math pedagogics with maths.
Learn the difference.

Secondly, no one can deny that scientists can be racists, and thus, by their personal feelings make the study/practice of science obnoxious to the targets of the scientist's prejudices.

This however, doesn't make statements like "math is racist" correct.

Rather, your obvious confusion is centered on the issue of this thread: To what extant is math a social construction vs. something that ontologically is?

The example of the African saying he has two dollars vs. a western accountant saying he has a negative net worth of $3 is meant to show a simple example. Indeed, it can be validly claimed that all numbers less than zero are a social construct. Yet, on a western test the African's answer will be wrong.

Similarly, Euclid's geometry is based upon six axioms. They are not proven. If you accept the axioms, then you may accept--in fact, cannot refute--all the rest of Euclidean geometry. However, Einstein had to reject these axioms in order to explain relativity. So is Euclidean geometry a social construct or does it exist ontologically?

Basic arithmetic, upon which all forms of math are built, has in its foundation certain commonly agreed-to, but nonetheless arbitrary, statements. If you accept these arbitrary statements, all current branches of math may be derived. But it is a social construct to agree to the arbitrary statements. So does basic arithmetic ontologically have being? If not, then all branches of math used to model reality are conveniences. Math does not exist as something that is, it exists by general agreement.

The question then becomes, which people were included in the "general" adjective in the above statement? Who agreed? What biases are introduced by those who are included vs. those who are not included?

Once again, nothing here intends to impute evil design in how these general agreements, in math and geometry for example, are reached. What does apply is the principle of Cohesiveness. If math is basically a social construct, and if everything about math, such as research, funding, who may critique, who may interpret, who may get assistance, who may be paid and how much, etc., is basically a human environment, then the possibility for bias in all social constructs around and within math become infinite. Therefore, all math and everything about it becomes open to scrutiny re bigotry.

All of the above applies to every branch of science as well. In fact, to the extent a science is fundamentally built upon math, it brings with it already-existing social constructs permitting bigotry from its initial formation.

In a different thread, someone asked the question, what is math to you? The consensus answer seemed to be, math is something that shows equality. However, the most common use of math is actually to show which amount is larger and which is lesser. The idea is that humans have preference for what is better and what is worse. More cows are better than less cows. More land to plant is better than less land to plant. More trade goods for my trade goods is better than the less trade goods someone else is offering. The arch supports more weight than a beam.

Exactly how we choose better and worse has become math. Math derives, then, from a social construct of a value judgement. If you think about it, you will see that the manner in which humans could have chosen to evolve a system symbolizing relationships could have taken a variety of forms. For example, nothing can change the ratio of the circumference of a circle to its diameter (pi). However, it is possible to envision a society which never made use of pi in any conscious way. In this case, does it matter if pi has ontological being?

 

[arildno]

"The example of the African saying he has two dollars vs. a western accountant saying he has a negative net worth of $3 is meant to show a simple example. Indeed, it can be validly claimed that all numbers less than zero are a social construct. Yet, on a western test the African's answer will be wrong."

By this, you have just shown that you don't know what you are talking about.
Learn the definitions of what we ordinarily call "negative", and, for that matter, what "ordering" as in "less" means.
You seem trapped in the concept of "zero as nothing", please learn some maths before posting anything further.

 

[owl3951]

True. And as far as I know, the moslems didn't do much with developing the subject anyway. I think Omar Khayyam worked a bit with cubic equations, maybe one or two other moslem mathematicians did some small pieces of original work. Nothing like the colossal European legacy. We have to remember the moslems primarily as transmitters rather than originators. Hindus, on the other hand, were more prolific.

The thing to remember about Islam is that it APPLIED so well what was known. There are two ways to excel at being a mathematical society: research and evolution of math, or use of the current understanding of math. Obviously, a combination of the two is lustrous.

What about differences between the races in aptitude and inclination?

Sigh. This is, not necessarily a difficult question, but a question requiring much response. Truthfully, I do not believe we know enough to answer accurately and fully the question. Altho there have been some efforts to derive culture-neutral measurements of both aptitude and ability, I am not happy with all of these instruments. Btw, I am a mere dilettante in this field. I have read some because I was curious, but I urge everyone to investigate on their own.

The first thought that comes with any statement regarding current aptitude and ability is that any true measure is normative. We cannot extrapolate how things might be if history were different, nor can we say much about future developments. If there is a statistically significant difference, do we know enough to explain WHY the difference exists? Will the difference always exist?

These questions apply whether we are measuring across race, gender or nationality.

The next question that comes to me is whether there would be a difference in applied vs. theoretical understandings across cultures. I guess I am making a somewhat arbitrary distinction between learning and scholarship. I view learning as being able to use well what is known and understood--even if that is a relatively small amount; while scholarship seeks to expand knowledge for knowledge's sake. Under scholarship, the idea is that application may someday catch up to the boundaries of what is now known.

Truthfully, for most people, learning geometry and algebra is scholarship. Most of the world makes due with simple arithmetic. In some ways, the ancient Greeks and Hindus are still way ahead, in a scholarly way, of the mass of humanity.

Finally, altho this idea is not PC, I have often wondered if there is not a species-survival rationale for the apparent differences in gender and race. I am thinking here of what economists call the Principle of Comparative Advantage. That is, if there are valid and continuing differences, is that because we are supposed to make use, somehow, in the seemingly random brownian motion of societal evolution, of the comparative strengths?

I am thinking now of the difference between a forest and a wheat field. The field, because it is a uniform species, requires much effort, fertilizer, insect and parasite control, etc. No one puts all this effort into a forest, because of diversification. Obviously, we and the planet need both agriculture and forests. So we once again bump up against the old balance, the yin and yang, etc., etc.

Take, for instance, all the posts in this thread about single-sex education. One of the outcomes of studying the differences in brains between male and female is the proposal that education might be most effective if boys and girls attended different schools. The genders appear to learn differently, as well as think differently. Maybe PC is our enemy, as regards the optimal evolution of society.

This is, of course, only speculation on my part. But I do see that as organisms evolve, specialization takes place. This is true not only of one organism in a species (organs, senses, etc.), but the functions of different members of the species (drones, soldiers, queen in a beehive). It is also true in organizations and administrations. It is true in research. It may be a universal truth.

If comparative strengths is a true principle, then it must be emphasized that we need not make a superior/inferior question re the specialist functions. The heart is not superior to the brain. The hive needs the soldiers and the queen. A business needs sales and production. A garden with a myriad colors is more breath-taking than a garden of all white roses.

 

[owl3951]

"The example of the African saying he has two dollars vs. a western accountant saying he has a negative net worth of $3 is meant to show a simple example. Indeed, it can be validly claimed that all numbers less than zero are a social construct. Yet, on a western test the African's answer will be wrong."

By this, you have just shown that you don't know what you are talking about.

Did you assume these are my ideas? How shallow must be your education.

Learn the definitions of what we ordinarily call "negative", and, for that matter, what "ordering" as in "less" means..

Not just shallow, but simplistic as well. The immature call upon some vague but vast "we" to be on their side in order to win arguments based upon emotional reaction.

You seem trapped in the concept of "zero as nothing", please learn some maths before posting anything further.

The day I need an abrasive and bombastic person such as yourself to give me permission to post will be the day they bury me.

 

[cragwolf]

If only one persons reads and understands INTELLECTUAL IMPOSTURES, after learning about it in this thread, then this thread will have provided enlightenment to at least one.

That's better. See, you're learning.

Note to cragwolf: I really liked your statement--'Sarcasm is only effective when you are sure about the truth of the topic in question.' Does this mean you believe TRUTH actually exists?

No, the quote means what it says. It does not mean or imply that I "believe TRUTH actually exists". So what did you like about my statement?

 

[houserichichi]

Some think that the subject that should be talked about is racism of the mathematiciasns and scientists, but I think that's not the subject because that is sort of clear to many people. What I think that should be talked about here is the racism of the math application: do we use western methods in math only? do we use math from other cultures?...

I'd actually go so far as to not include that when defining what racist mathematics would be. To me, whether we use math from a French mathematician or that of an obscure Indian one is irrelevant, and I think any true mathematician would agree (no offense intended to anyone, of course). It's all dependent on who comes up with the best idea first (and through Ramanujan we see that it's not necessary to be published initially either). We'd all be taking Ramanujan just as seriously whether he was Indian, Egyptian, or Canadian - fact of the matter is the man earned his place in math history through his mind, not his mouth or opinions.

 

[<<<GUILLE>>>]

I wrote that post so that people didn't cary on believing I was extremist in this subject. I'm so anti-extremism that when I'm too extremist about being anti-extremist I stop being it for being against it.

In reallty I think like you.

 

[bombadillo]

Basically, Lacan equates "the exuberance of wild imagination" (his interpretation of the imaginary unit) with the wild, unfettered joy as symbolized by the phallus.

I know, it IS stupid; one of the few utterances which tops it is L. Irigaray's opinion that the main reason why the mechanics of solids (study of beams and rigid constructions and the like) has been more successful than fluid mechanics, is that beams are a phallic, masculine symbol, whereas fluids as in menstrual blood, is a symbol of the feminine.

Thus, due to the evils of patriarchy, the science of fluid mechanics has been held down by the science of solids.

What people like Lacan, Kristeva, and others write is complete drivel. If you want to generate your own postmodernist gobbledygook, go to:

http://www.elsewhere.org/cgi-bin/postmodern/

It will have you in fits.

Rather, your obvious confusion is centered on the issue of this thread: To what extant is math a social construction vs. something that ontologically is?

The example of the African saying he has two dollars vs. a western accountant saying he has a negative net worth of $3 is meant to show a simple example. Indeed, it can be validly claimed that all numbers less than zero are a social construct. Yet, on a western test the African's answer will be wrong.

Similarly, Euclid's geometry is based upon six axioms. They are not proven. If you accept the axioms, then you may accept--in fact, cannot refute--all the rest of Euclidean geometry. However, Einstein had to reject these axioms in order to explain relativity. So is Euclidean geometry a social construct or does it exist ontologically?

Ah yes, "socially constructed." Another buzzword much in vogue with the po-mo crowd. If a po-mo ("postmodernist") listens to a talk, say, on Quantum Field Theory, and understands not one equation, at least he can say, " A quantum field is socially constructed." The point is: everything is "socially constructed"; nothing exists "ontologically"; go and read Kant's Critique of Pure Reason: we can never come to grips with the thing-in-itself. Questions of what exists ontologically are passe; they reflect ignorance of Western philosophy. +3 doesn't have any greater ontological reality than -4: both are constructs.

Einstein didn't "reject" Euclidean axioms: you really have to go and learn some serious mathematics and physics. Euclidean geometry is a particular kind of Riemannian geometry, and it's the latter that is used in GR. As a matter of fact, I don't think Einstein even thought about the geometry initially: he was working with tensor equations. People like Minkowski provided a geometric interpretation.

 

[owl3951]

Tch. So Many "Shoulds"

There are a lot of people telling other people they "should" do things in these forums. The use of the parental stance seems to give weight to what the scolding person is saying. It implies the scolder is an authority, and the viewpoints they give are the correct and universally accepted views. Generally "shoulds" are an emotional reaction to what the scoldee has said.

Subsequently, I auto-reject scolders as being too lazy to think about how to correctly present the views they hold, and too emotional to describe proper reasoning. Scolders views may, in fact, be correct, but the presentation does not so demonstrate that correctness.

Ah yes, "socially constructed." Another buzzword much in vogue with the po-mo crowd...

Here is my first example: by a presentation of universal disdain for a class of "thinkers" called "the po-mo crowd" one seems to take an elevated and obviously more learned position. I admit I am not much of a follower of post-modernists. Nevertheless, I need not resort to scathing condemnation to reply to anything a "po-mo" stipulates. Nor does the rest of your response vindicate any sort of elevated learning.

It is irrelevant which vocabulary terms are used to describe the fact that math is a social construct. In 1921 Einstein gave a lecture in Berlin that basically says the same thing. He used the vocabulary of the time. In essence he demonstrates that math is detached from the world it pretends to describe. Math is a system of axioms and theories that are self-consistent, but not necessarily consistent with anything else. You will find a hyperlink to a translation of this lecture in the attached document.

So by condemning the vocabulary you appear to invalidate the statements made based upon it. The vocabulary, and its source, is irrelevant. The concepts described by the vocabulary could be translated to almost infinite systems of vocabulary and description.

You then, in fact, validate the concepts with this next sentence.

The point is: everything is "socially constructed"; nothing exists "ontologically"...

However, you then make a sweeping generalization that you state is universally accepted by all Western philosophers, and once again, seem to set yourself up as an authority with the right to be a parent to others.

go and read Kant's Critique of Pure Reason: we can never come to grips with the thing-in-itself. Questions of what exists ontologically are passe; they reflect ignorance of Western

Many people have critiqued Kant's Critque. Here is a nice one, if you want to see it.

personal.unizd.hr/~mjakic/data/kant.pdf

But there are tons more. I feel comfortable rejecting Kant as the infallible last word on the subject. Many people do. You can discover more by Googling.

You then go on to make a point I had already made, as if somehow reiteration is meaningful to something you are trying to refute.

+3 doesn't have any greater ontological reality than -4: both are constructs.

I had already stipulated that an almost infinite series of symbologies could have arisen in the original thinking of specifying greater and lesser. If you wonder why I stated negative numbers were a social construct, it is because I wanted to avoid the arument that, "I see one apple. There is one. The apple exists. Therefore 'one' exists, because I can define a cardinality between "number" and "apple". I see two apples..." and so on. I knew if I said anything to reflect on the existential reality of this concept, I would have to do more typing.

Einstein didn't "reject" Euclidean axioms: you really have to go and learn some serious mathematics and physics. Euclidean geometry is a particular kind of Riemannian geometry, and it's the latter that is used in GR. As a matter of fact, I don't think Einstein even thought about the geometry initially: he was working with tensor equations. People like Minkowski provided a geometric interpretation.

And here you leave me completely baffled. Do you know ANYTHING at all about what you are declaiming here, as if you are expert? Relativity is nothing BUT a theory of the geometry of space. Geometry pre-occupied Einstein. There are innumerable quotes from him on this whole topic. He gave lectures about what geometry really is. He spoke a variety of times about how much more difficult the development of relativity was because of the struggle with abandoning the "magnificent" and "beautiful" Euclidean geometry. The curvature tensor you speak of is a descriptor of behavior in a specific Reimann geometry (hyperbolic). If you are implying Einstein didn't know this, I am dumbfounded. If you are implying he put himself to the trouble to learn tensor calculus because he just wanted to, I do not know what to say. The whole point of him learning tensor calculus was because he realized he needed tools to work with a non-Euclidean geometry.

The attached document contains some quotes from Einstein on the subject. It also has a partial summary of some of his derived outcomes of relativity in regards to geometry. It has the link to that lecture in Berlin I mentioned above. I could have made the attached document more voluminous to make my point. But the contents should be sufficient as is.

It is no wonder teenagers eventually reject the heavy-handed, autocratic parent who declares: "Because I said so!"

Hmmm...when I click on "manage attachments" nothing happens. Si O woll cut and paste a few ezamples directly here.

From the latest results of the theory of relativity it is probable that our three dimensional space is also approximately spherical, that is, that the laws of disposition of rigid bodies in it are not given by Euclidean geometry but approximately by spherical geometry. (Albert Einstein, 1954)

Special relativity is still based directly on an empirical law, that of the constancy of the velocity of light. dx2 + dy2 + dz2 =(ct)2 where ct is the distance traveled by light c in time t. The fact that such a metric is called Euclidean is connected with the following. The postulation of such a metric in a three dimensional continuum is fully equivalent to the postulation of the axioms of Euclidean Geometry. The defining equation of the metric is then nothing but the Pythagorean theorem applied to the differentials of the co-ordinates. … In the special theory of relativity those co-ordinate changes (by transformation) are permitted for which also in the new co-ordinate system the quantity (ct)2 equals the sum of the squares of the co-ordinate differentials. Such transformations are called Lorentz transformations. (Einstein, 1954)

The defining equation of the metric is then nothing but the Pythagorean theorem applied to the differentials of the co-ordinates. (Albert Einstein, 1954)

'But the path (of general relativity) was thornier than one might suppose, because it demanded the abandonment of Euclidean geometry. This is what we mean when we talk of the 'curvature of space'. The fundamental concepts of the 'straight line', the 'plane', etc., thereby lose their precise significance in physics.
In the general theory of relativity the doctrine of space and time, or kinematics, no longer figures as a fundamental independent of the rest of physics. The geometrical behavior of bodies and the motion of clocks rather depend on gravitational fields which in their turn are produced by matter.' (Albert Einstein, 1919)

If Euclid failed to kindle your youthful enthusiasm, then you were not born to be a scientific thinker. A. Einstein: Zur Methodik der theoretischen Physik. In: Mein Weltbild.

"In your schooldays most of you who read this book made acquaintance with the noble building of Euclid's geometry, and you remember--perhaps with more respect than love--the magnificent structure, on the lofty staircase of which you were chased about for uncounted hours by conscientious teachers…Assuredly by force of this bit of your past you would beat with contempt anyone who casts doubts on even the most out of the way fragment of any of its propositions." Relativity: The Special and General Theory" (Crown, 1961)

SUMMARY OF SOME OF EINSTEIN’S IMPORTANT FINDINGS
· Gravitation is not a force
· Physics = Geometry of space-time
· Gravitation = space-time curvature
· Relativity theory is ultimately about the nature of gravitation
· Relativity explains gravitation in terms of curved space-time, i.e. Geometry
· "Gravitational force" becomes an effect of the geometry of space-time
· The curvature of space-time is measured by a "curvature tensor" (Riemann's geometry)
· Each point is described by ten numbers (metric tensor)
· Euclid's geometry is one of the infinite possible metric tensors (zero curvature)
· Other geometries describe spaces that are not flat, but have warps
· What causes the "warps" is energy-mass
· Clocks slow down in a gravitational field
· Light is deflected in a gravitational field

http://www.tu-harburg.de/rzt/rzt/it/Geometry.html